2-ancestry mixing, positive discriminant

Percentage Accurate: 43.4% → 95.8%

Time: 12.0s

Alternatives: 10

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Initial Program: 43.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 95.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}\right)}^{-1} + \sqrt[3]{-g} \cdot {\left(\sqrt[3]{a}\right)}^{-1} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (pow (cbrt (/ (* -2.0 a) (- g (* (- g) -1.0)))) -1.0)
  (* (cbrt (- g)) (pow (cbrt a) -1.0))))

C

double code(double g, double h, double a) {
	return pow(cbrt(((-2.0 * a) / (g - (-g * -1.0)))), -1.0) + (cbrt(-g) * pow(cbrt(a), -1.0));
}

Java

public static double code(double g, double h, double a) {
	return Math.pow(Math.cbrt(((-2.0 * a) / (g - (-g * -1.0)))), -1.0) + (Math.cbrt(-g) * Math.pow(Math.cbrt(a), -1.0));
}

Julia

function code(g, h, a)
	return Float64((cbrt(Float64(Float64(-2.0 * a) / Float64(g - Float64(Float64(-g) * -1.0)))) ^ -1.0) + Float64(cbrt(Float64(-g)) * (cbrt(a) ^ -1.0)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[Power[N[(N[(-2.0 * a), $MachinePrecision] / N[(g - N[((-g) * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], -1.0], $MachinePrecision] + N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[N[Power[a, 1/3], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

4. Applied rewrites 40.5%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

5. Taylor expanded in g around -inf

   \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{-1 \cdot \left(g \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-1 \cdot g\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(\mathsf{neg}\left(g\right)\right)} \cdot {\left(\sqrt{-1}\right)}^{2}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(\mathsf{neg}\left(g\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   4. rem-square-sqrt N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(\mathsf{neg}\left(g\right)\right) \cdot \color{blue}{-1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(\mathsf{neg}\left(g\right)\right) \cdot -1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. lower-neg.f64 23.1

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-g\right)} \cdot -1}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

7. Applied rewrites 23.1%

   \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-g\right) \cdot -1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

9. Applied rewrites 45.2%

   \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \color{blue}{\sqrt[3]{\left(-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)\right) \cdot 0.5} \cdot {\left(\sqrt[3]{a}\right)}^{-1}} \]

10. Taylor expanded in g around inf

    \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{-1 \cdot g}} \cdot {\left(\sqrt[3]{a}\right)}^{-1} \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\mathsf{neg}\left(g\right)}} \cdot {\left(\sqrt[3]{a}\right)}^{-1} \]

    2. lower-neg.f64 95.9

       \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{-g}} \cdot {\left(\sqrt[3]{a}\right)}^{-1} \]

12. Applied rewrites 95.9%

    \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{-g}} \cdot {\left(\sqrt[3]{a}\right)}^{-1} \]

13. Final simplification 95.9%

    \[\leadsto {\left(\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}\right)}^{-1} + \sqrt[3]{-g} \cdot {\left(\sqrt[3]{a}\right)}^{-1} \]
14. Add Preprocessing

Alternative 2: 87.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-g}{a}}\\ \mathbf{if}\;g \leq -3.65 \cdot 10^{+152}:\\ \;\;\;\;t\_0 + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}\\ \mathbf{elif}\;g \leq -4.8 \cdot 10^{-138}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}\right)}^{-1} + \sqrt[3]{\frac{\left(-g\right) - \left(-g\right)}{a \cdot 2}}\\ \mathbf{elif}\;g \leq 1.2 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\sqrt[3]{a}} + t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}\right)}^{-1} + \frac{\sqrt[3]{-0.5 \cdot \mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}{\sqrt[3]{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ (- g) a))))
   (if (<= g -3.65e+152)
     (+ t_0 (cbrt (* -0.25 (* (/ h g) (/ h a)))))
     (if (<= g -4.8e-138)
       (+
        (pow
         (* (cbrt (/ -1.0 (- (sqrt (* (+ h g) (- g h))) g))) (cbrt (* -2.0 a)))
         -1.0)
        (cbrt (/ (- (- g) (- g)) (* a 2.0))))
       (if (<= g 1.2e-264)
         (+ (/ (cbrt (* (* -0.25 (/ h g)) h)) (cbrt a)) t_0)
         (+
          (pow (cbrt (/ (* -2.0 a) (- g (* (- g) -1.0)))) -1.0)
          (/
           (cbrt (* -0.5 (fma (sqrt (+ h g)) (sqrt (- g h)) g)))
           (cbrt a))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((-g / a));
	double tmp;
	if (g <= -3.65e+152) {
		tmp = t_0 + cbrt((-0.25 * ((h / g) * (h / a))));
	} else if (g <= -4.8e-138) {
		tmp = pow((cbrt((-1.0 / (sqrt(((h + g) * (g - h))) - g))) * cbrt((-2.0 * a))), -1.0) + cbrt(((-g - -g) / (a * 2.0)));
	} else if (g <= 1.2e-264) {
		tmp = (cbrt(((-0.25 * (h / g)) * h)) / cbrt(a)) + t_0;
	} else {
		tmp = pow(cbrt(((-2.0 * a) / (g - (-g * -1.0)))), -1.0) + (cbrt((-0.5 * fma(sqrt((h + g)), sqrt((g - h)), g))) / cbrt(a));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(-g) / a))
	tmp = 0.0
	if (g <= -3.65e+152)
		tmp = Float64(t_0 + cbrt(Float64(-0.25 * Float64(Float64(h / g) * Float64(h / a)))));
	elseif (g <= -4.8e-138)
		tmp = Float64((Float64(cbrt(Float64(-1.0 / Float64(sqrt(Float64(Float64(h + g) * Float64(g - h))) - g))) * cbrt(Float64(-2.0 * a))) ^ -1.0) + cbrt(Float64(Float64(Float64(-g) - Float64(-g)) / Float64(a * 2.0))));
	elseif (g <= 1.2e-264)
		tmp = Float64(Float64(cbrt(Float64(Float64(-0.25 * Float64(h / g)) * h)) / cbrt(a)) + t_0);
	else
		tmp = Float64((cbrt(Float64(Float64(-2.0 * a) / Float64(g - Float64(Float64(-g) * -1.0)))) ^ -1.0) + Float64(cbrt(Float64(-0.5 * fma(sqrt(Float64(h + g)), sqrt(Float64(g - h)), g))) / cbrt(a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, -3.65e+152], N[(t$95$0 + N[Power[N[(-0.25 * N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -4.8e-138], N[(N[Power[N[(N[Power[N[(-1.0 / N[(N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-2.0 * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] + N[Power[N[(N[((-g) - (-g)), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1.2e-264], N[(N[(N[Power[N[(N[(-0.25 * N[(h / g), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[Power[N[Power[N[(N[(-2.0 * a), $MachinePrecision] / N[(g - N[((-g) * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], -1.0], $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if g < -3.6500000000000002e152

   1. Initial program 1.5%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

      4. lower-neg.f64 3.6

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

   5. Applied rewrites 3.6%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      2. lower-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{h \cdot h}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      4. times-frac N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a}} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \color{blue}{\frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \color{blue}{\left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\color{blue}{\sqrt[3]{\frac{-1}{2}}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      10. lower-cbrt.f64 64.0

          \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{\sqrt[3]{0.5}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   8. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} \]

      3. lower-+.f64 64.0

         \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} \]

   10. Applied rewrites 64.0%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}} \]

   if -3.6500000000000002e152 < g < -4.7999999999999998e-138

   1. Initial program 77.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. Applied rewrites 97.7%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   7. Taylor expanded in g around -inf

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{-1 \cdot g}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(\mathsf{neg}\left(g\right)\right)}\right)} \]

      2. lower-neg.f64 97.7

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(-g\right)}\right)} \]

   9. Applied rewrites 97.7%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(-g\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \left(-g\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\color{blue}{\left(\left(-g\right) - \left(-g\right)\right) \cdot \frac{1}{2 \cdot a}}} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \color{blue}{\frac{1}{2 \cdot a}}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \frac{1}{\color{blue}{2 \cdot a}}} \]

       5. associate-/r* N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \frac{\color{blue}{\frac{1}{2}}}{a}} \]

       7. clear-num N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \color{blue}{\frac{1}{\frac{a}{\frac{1}{2}}}}} \]

       8. un-div-inv N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{\left(-g\right) - \left(-g\right)}{\frac{a}{\frac{1}{2}}}}} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{\left(-g\right) - \left(-g\right)}{\frac{a}{\frac{1}{2}}}}} \]

   11. Applied rewrites 97.7%

       \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{\left(-g\right) - \left(-g\right)}{a \cdot 2}}} \]

   if -4.7999999999999998e-138 < g < 1.1999999999999999e-264

   1. Initial program 42.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

      4. lower-neg.f64 15.8

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

   5. Applied rewrites 15.8%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      2. lower-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{h \cdot h}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      4. times-frac N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a}} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \color{blue}{\frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \color{blue}{\left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\color{blue}{\sqrt[3]{\frac{-1}{2}}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      10. lower-cbrt.f64 74.3

          \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{\sqrt[3]{0.5}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   8. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.4%

       \[\leadsto \sqrt[3]{\frac{h}{a}} \cdot \color{blue}{\sqrt[3]{\frac{h}{g} \cdot -0.25}} + \sqrt[3]{\frac{-g}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 75.4%

       \[\leadsto \frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\color{blue}{\sqrt[3]{a}}} + \sqrt[3]{\frac{-g}{a}} \]

   if 1.1999999999999999e-264 < g

   1. Initial program 44.7%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   5. Taylor expanded in g around -inf

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{-1 \cdot \left(g \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-1 \cdot g\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(\mathsf{neg}\left(g\right)\right)} \cdot {\left(\sqrt{-1}\right)}^{2}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(\mathsf{neg}\left(g\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      4. rem-square-sqrt N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(\mathsf{neg}\left(g\right)\right) \cdot \color{blue}{-1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(\mathsf{neg}\left(g\right)\right) \cdot -1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      6. lower-neg.f64 46.3

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-g\right)} \cdot -1}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   7. Applied rewrites 46.3%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-g\right) \cdot -1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   9. Applied rewrites 96.2%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \color{blue}{\sqrt[3]{\left(-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)\right) \cdot 0.5} \cdot {\left(\sqrt[3]{a}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \color{blue}{\sqrt[3]{\left(-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)\right) \cdot \frac{1}{2}} \cdot {\left(\sqrt[3]{a}\right)}^{-1}} \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\left(-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)\right) \cdot \frac{1}{2}} \cdot \color{blue}{{\left(\sqrt[3]{a}\right)}^{-1}} \]

       3. unpow-1 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\left(-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)\right) \cdot \frac{1}{2}} \cdot \color{blue}{\frac{1}{\sqrt[3]{a}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \color{blue}{\frac{\sqrt[3]{\left(-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)\right) \cdot \frac{1}{2}}}{\sqrt[3]{a}}} \]

       5. lower-/.f64 96.2

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \color{blue}{\frac{\sqrt[3]{\left(-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)\right) \cdot 0.5}}{\sqrt[3]{a}}} \]

   11. Applied rewrites 96.2%

       \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \color{blue}{\frac{\sqrt[3]{-0.5 \cdot \mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}{\sqrt[3]{a}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -3.65 \cdot 10^{+152}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}\\ \mathbf{elif}\;g \leq -4.8 \cdot 10^{-138}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}\right)}^{-1} + \sqrt[3]{\frac{\left(-g\right) - \left(-g\right)}{a \cdot 2}}\\ \mathbf{elif}\;g \leq 1.2 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}\right)}^{-1} + \frac{\sqrt[3]{-0.5 \cdot \mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}}{\sqrt[3]{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-g}{a}}\\ \mathbf{if}\;g \leq -3.65 \cdot 10^{+152}:\\ \;\;\;\;t\_0 + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}\\ \mathbf{elif}\;g \leq -4.8 \cdot 10^{-138}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}\right)}^{-1} + \sqrt[3]{\frac{\left(-g\right) - \left(-g\right)}{a \cdot 2}}\\ \mathbf{elif}\;g \leq 1.2 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\sqrt[3]{a}} + t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{g - g}{-2}}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ (- g) a))))
   (if (<= g -3.65e+152)
     (+ t_0 (cbrt (* -0.25 (* (/ h g) (/ h a)))))
     (if (<= g -4.8e-138)
       (+
        (pow
         (* (cbrt (/ -1.0 (- (sqrt (* (+ h g) (- g h))) g))) (cbrt (* -2.0 a)))
         -1.0)
        (cbrt (/ (- (- g) (- g)) (* a 2.0))))
       (if (<= g 1.2e-264)
         (+ (/ (cbrt (* (* -0.25 (/ h g)) h)) (cbrt a)) t_0)
         (fma
          (cbrt (/ 0.5 a))
          (cbrt (- (fma (sqrt (+ h g)) (sqrt (- g h)) g)))
          (cbrt (/ (/ (- g g) -2.0) a))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((-g / a));
	double tmp;
	if (g <= -3.65e+152) {
		tmp = t_0 + cbrt((-0.25 * ((h / g) * (h / a))));
	} else if (g <= -4.8e-138) {
		tmp = pow((cbrt((-1.0 / (sqrt(((h + g) * (g - h))) - g))) * cbrt((-2.0 * a))), -1.0) + cbrt(((-g - -g) / (a * 2.0)));
	} else if (g <= 1.2e-264) {
		tmp = (cbrt(((-0.25 * (h / g)) * h)) / cbrt(a)) + t_0;
	} else {
		tmp = fma(cbrt((0.5 / a)), cbrt(-fma(sqrt((h + g)), sqrt((g - h)), g)), cbrt((((g - g) / -2.0) / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(-g) / a))
	tmp = 0.0
	if (g <= -3.65e+152)
		tmp = Float64(t_0 + cbrt(Float64(-0.25 * Float64(Float64(h / g) * Float64(h / a)))));
	elseif (g <= -4.8e-138)
		tmp = Float64((Float64(cbrt(Float64(-1.0 / Float64(sqrt(Float64(Float64(h + g) * Float64(g - h))) - g))) * cbrt(Float64(-2.0 * a))) ^ -1.0) + cbrt(Float64(Float64(Float64(-g) - Float64(-g)) / Float64(a * 2.0))));
	elseif (g <= 1.2e-264)
		tmp = Float64(Float64(cbrt(Float64(Float64(-0.25 * Float64(h / g)) * h)) / cbrt(a)) + t_0);
	else
		tmp = fma(cbrt(Float64(0.5 / a)), cbrt(Float64(-fma(sqrt(Float64(h + g)), sqrt(Float64(g - h)), g))), cbrt(Float64(Float64(Float64(g - g) / -2.0) / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, -3.65e+152], N[(t$95$0 + N[Power[N[(-0.25 * N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -4.8e-138], N[(N[Power[N[(N[Power[N[(-1.0 / N[(N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-2.0 * a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] + N[Power[N[(N[((-g) - (-g)), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1.2e-264], N[(N[(N[Power[N[(N[(-0.25 * N[(h / g), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-N[(N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] + N[Power[N[(N[(N[(g - g), $MachinePrecision] / -2.0), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if g < -3.6500000000000002e152

   1. Initial program 1.5%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

      4. lower-neg.f64 3.6

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

   5. Applied rewrites 3.6%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      2. lower-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{h \cdot h}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      4. times-frac N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a}} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \color{blue}{\frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \color{blue}{\left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\color{blue}{\sqrt[3]{\frac{-1}{2}}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      10. lower-cbrt.f64 64.0

          \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{\sqrt[3]{0.5}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   8. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} \]

      3. lower-+.f64 64.0

         \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} \]

   10. Applied rewrites 64.0%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}} \]

   if -3.6500000000000002e152 < g < -4.7999999999999998e-138

   1. Initial program 77.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. Applied rewrites 97.7%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   7. Taylor expanded in g around -inf

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{-1 \cdot g}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(\mathsf{neg}\left(g\right)\right)}\right)} \]

      2. lower-neg.f64 97.7

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(-g\right)}\right)} \]

   9. Applied rewrites 97.7%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(-g\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \left(-g\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\color{blue}{\left(\left(-g\right) - \left(-g\right)\right) \cdot \frac{1}{2 \cdot a}}} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \color{blue}{\frac{1}{2 \cdot a}}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \frac{1}{\color{blue}{2 \cdot a}}} \]

       5. associate-/r* N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \frac{\color{blue}{\frac{1}{2}}}{a}} \]

       7. clear-num N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \color{blue}{\frac{1}{\frac{a}{\frac{1}{2}}}}} \]

       8. un-div-inv N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{\left(-g\right) - \left(-g\right)}{\frac{a}{\frac{1}{2}}}}} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{\left(-g\right) - \left(-g\right)}{\frac{a}{\frac{1}{2}}}}} \]

   11. Applied rewrites 97.7%

       \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{\left(-g\right) - \left(-g\right)}{a \cdot 2}}} \]

   if -4.7999999999999998e-138 < g < 1.1999999999999999e-264

   1. Initial program 42.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

      4. lower-neg.f64 15.8

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

   5. Applied rewrites 15.8%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      2. lower-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{h \cdot h}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      4. times-frac N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a}} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \color{blue}{\frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \color{blue}{\left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\color{blue}{\sqrt[3]{\frac{-1}{2}}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      10. lower-cbrt.f64 74.3

          \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{\sqrt[3]{0.5}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   8. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.4%

       \[\leadsto \sqrt[3]{\frac{h}{a}} \cdot \color{blue}{\sqrt[3]{\frac{h}{g} \cdot -0.25}} + \sqrt[3]{\frac{-g}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 75.4%

       \[\leadsto \frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\color{blue}{\sqrt[3]{a}}} + \sqrt[3]{\frac{-g}{a}} \]

   if 1.1999999999999999e-264 < g

   1. Initial program 44.7%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   5. Taylor expanded in g around -inf

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{-1 \cdot \left(g \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-1 \cdot g\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(\mathsf{neg}\left(g\right)\right)} \cdot {\left(\sqrt{-1}\right)}^{2}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(\mathsf{neg}\left(g\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      4. rem-square-sqrt N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(\mathsf{neg}\left(g\right)\right) \cdot \color{blue}{-1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(\mathsf{neg}\left(g\right)\right) \cdot -1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      6. lower-neg.f64 46.3

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-g\right)} \cdot -1}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   7. Applied rewrites 46.3%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-g\right) \cdot -1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a}} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}{2 \cdot a}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\color{blue}{2 \cdot a}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\color{blue}{a \cdot 2}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{a \cdot \color{blue}{\frac{2}{1}}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\color{blue}{\frac{a \cdot 2}{1}}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\frac{\color{blue}{2 \cdot a}}{1}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\frac{\color{blue}{2 \cdot a}}{1}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\frac{2 \cdot a}{1}}{\color{blue}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\color{blue}{\frac{\frac{2 \cdot a}{1}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\frac{\color{blue}{2 \cdot a}}{1}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\frac{\color{blue}{a \cdot 2}}{1}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}} \]

      16. associate-/l* N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\color{blue}{a \cdot \frac{2}{1}}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{a \cdot \color{blue}{2}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}} \]

      18. lower-*.f64 46.3

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}} \]

      19. lift--.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{a \cdot 2}{\color{blue}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}}} \]

      20. lift--.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{a \cdot 2}{\left(-g\right) - \sqrt{\color{blue}{g \cdot g - h \cdot h}}}}} \]

   9. Applied rewrites 68.8%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1}{\frac{a \cdot 2}{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}}}} \]

   11. Applied rewrites 96.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{g - g}{-2}}{a}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -3.65 \cdot 10^{+152}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}\\ \mathbf{elif}\;g \leq -4.8 \cdot 10^{-138}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}\right)}^{-1} + \sqrt[3]{\frac{\left(-g\right) - \left(-g\right)}{a \cdot 2}}\\ \mathbf{elif}\;g \leq 1.2 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{g - g}{-2}}{a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a}}\\ t_1 := \sqrt[3]{\frac{-g}{a}}\\ \mathbf{if}\;g \leq -2.15 \cdot 10^{+151}:\\ \;\;\;\;t\_1 + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}\\ \mathbf{elif}\;g \leq -4.8 \cdot 10^{-138}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}, \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \frac{0.5}{a}}\right)\\ \mathbf{elif}\;g \leq 1.2 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\sqrt[3]{a}} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{g - g}{-2}}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ 0.5 a))) (t_1 (cbrt (/ (- g) a))))
   (if (<= g -2.15e+151)
     (+ t_1 (cbrt (* -0.25 (* (/ h g) (/ h a)))))
     (if (<= g -4.8e-138)
       (fma
        t_0
        (cbrt (- (sqrt (* (+ h g) (- g h))) g))
        (cbrt (* (- (- g) (- g)) (/ 0.5 a))))
       (if (<= g 1.2e-264)
         (+ (/ (cbrt (* (* -0.25 (/ h g)) h)) (cbrt a)) t_1)
         (fma
          t_0
          (cbrt (- (fma (sqrt (+ h g)) (sqrt (- g h)) g)))
          (cbrt (/ (/ (- g g) -2.0) a))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((0.5 / a));
	double t_1 = cbrt((-g / a));
	double tmp;
	if (g <= -2.15e+151) {
		tmp = t_1 + cbrt((-0.25 * ((h / g) * (h / a))));
	} else if (g <= -4.8e-138) {
		tmp = fma(t_0, cbrt((sqrt(((h + g) * (g - h))) - g)), cbrt(((-g - -g) * (0.5 / a))));
	} else if (g <= 1.2e-264) {
		tmp = (cbrt(((-0.25 * (h / g)) * h)) / cbrt(a)) + t_1;
	} else {
		tmp = fma(t_0, cbrt(-fma(sqrt((h + g)), sqrt((g - h)), g)), cbrt((((g - g) / -2.0) / a)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(0.5 / a))
	t_1 = cbrt(Float64(Float64(-g) / a))
	tmp = 0.0
	if (g <= -2.15e+151)
		tmp = Float64(t_1 + cbrt(Float64(-0.25 * Float64(Float64(h / g) * Float64(h / a)))));
	elseif (g <= -4.8e-138)
		tmp = fma(t_0, cbrt(Float64(sqrt(Float64(Float64(h + g) * Float64(g - h))) - g)), cbrt(Float64(Float64(Float64(-g) - Float64(-g)) * Float64(0.5 / a))));
	elseif (g <= 1.2e-264)
		tmp = Float64(Float64(cbrt(Float64(Float64(-0.25 * Float64(h / g)) * h)) / cbrt(a)) + t_1);
	else
		tmp = fma(t_0, cbrt(Float64(-fma(sqrt(Float64(h + g)), sqrt(Float64(g - h)), g))), cbrt(Float64(Float64(Float64(g - g) / -2.0) / a)));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, -2.15e+151], N[(t$95$1 + N[Power[N[(-0.25 * N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -4.8e-138], N[(t$95$0 * N[Power[N[(N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[((-g) - (-g)), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, 1.2e-264], N[(N[(N[Power[N[(N[(-0.25 * N[(h / g), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$0 * N[Power[(-N[(N[Sqrt[N[(h + g), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(g - h), $MachinePrecision]], $MachinePrecision] + g), $MachinePrecision]), 1/3], $MachinePrecision] + N[Power[N[(N[(N[(g - g), $MachinePrecision] / -2.0), $MachinePrecision] / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if g < -2.14999999999999991e151

   1. Initial program 3.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

      4. lower-neg.f64 3.9

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

   5. Applied rewrites 3.9%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      2. lower-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{h \cdot h}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      4. times-frac N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a}} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \color{blue}{\frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \color{blue}{\left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\color{blue}{\sqrt[3]{\frac{-1}{2}}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      10. lower-cbrt.f64 64.5

          \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{\sqrt[3]{0.5}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   8. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} \]

      3. lower-+.f64 64.5

         \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} \]

   10. Applied rewrites 64.5%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}} \]

   if -2.14999999999999991e151 < g < -4.7999999999999998e-138

   1. Initial program 76.6%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. Applied rewrites 97.6%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   7. Taylor expanded in g around -inf

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{-1 \cdot g}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(\mathsf{neg}\left(g\right)\right)}\right)} \]

      2. lower-neg.f64 97.6

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(-g\right)}\right)} \]

   9. Applied rewrites 97.6%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(-g\right)}\right)} \]

   11. Applied rewrites 97.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}, \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \frac{0.5}{a}}\right)} \]

   if -4.7999999999999998e-138 < g < 1.1999999999999999e-264

   1. Initial program 42.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

      4. lower-neg.f64 15.8

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

   5. Applied rewrites 15.8%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      2. lower-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{h \cdot h}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      4. times-frac N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a}} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \color{blue}{\frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \color{blue}{\left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\color{blue}{\sqrt[3]{\frac{-1}{2}}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      10. lower-cbrt.f64 74.3

          \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{\sqrt[3]{0.5}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   8. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.4%

       \[\leadsto \sqrt[3]{\frac{h}{a}} \cdot \color{blue}{\sqrt[3]{\frac{h}{g} \cdot -0.25}} + \sqrt[3]{\frac{-g}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 75.4%

       \[\leadsto \frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\color{blue}{\sqrt[3]{a}}} + \sqrt[3]{\frac{-g}{a}} \]

   if 1.1999999999999999e-264 < g

   1. Initial program 44.7%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   5. Taylor expanded in g around -inf

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{-1 \cdot \left(g \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-1 \cdot g\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(\mathsf{neg}\left(g\right)\right)} \cdot {\left(\sqrt{-1}\right)}^{2}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(\mathsf{neg}\left(g\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      4. rem-square-sqrt N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(\mathsf{neg}\left(g\right)\right) \cdot \color{blue}{-1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(\mathsf{neg}\left(g\right)\right) \cdot -1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      6. lower-neg.f64 46.3

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-g\right)} \cdot -1}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   7. Applied rewrites 46.3%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \color{blue}{\left(-g\right) \cdot -1}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a}} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}{2 \cdot a}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\color{blue}{2 \cdot a}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\color{blue}{a \cdot 2}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{a \cdot \color{blue}{\frac{2}{1}}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\color{blue}{\frac{a \cdot 2}{1}}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\frac{\color{blue}{2 \cdot a}}{1}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\frac{\color{blue}{2 \cdot a}}{1}}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}} \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\frac{2 \cdot a}{1}}{\color{blue}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\color{blue}{\frac{\frac{2 \cdot a}{1}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\frac{\color{blue}{2 \cdot a}}{1}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\frac{\color{blue}{a \cdot 2}}{1}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}} \]

      16. associate-/l* N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\color{blue}{a \cdot \frac{2}{1}}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{a \cdot \color{blue}{2}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}} \]

      18. lower-*.f64 46.3

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}} \]

      19. lift--.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{a \cdot 2}{\color{blue}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}}} \]

      20. lift--.f64 N/A

          \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\frac{1}{\frac{a \cdot 2}{\left(-g\right) - \sqrt{\color{blue}{g \cdot g - h \cdot h}}}}} \]

   9. Applied rewrites 68.8%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \left(-g\right) \cdot -1}}} + \sqrt[3]{\color{blue}{\frac{1}{\frac{a \cdot 2}{-\mathsf{fma}\left(\sqrt{g - h}, \sqrt{h + g}, g\right)}}}} \]

   11. Applied rewrites 96.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{-\mathsf{fma}\left(\sqrt{h + g}, \sqrt{g - h}, g\right)}, \sqrt[3]{\frac{\frac{g - g}{-2}}{a}}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-g}{a}}\\ \mathbf{if}\;g \leq -2.15 \cdot 10^{+151}:\\ \;\;\;\;t\_0 + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}\\ \mathbf{elif}\;g \leq -4.8 \cdot 10^{-138}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}, \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \frac{0.5}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\sqrt[3]{a}} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (/ (- g) a))))
   (if (<= g -2.15e+151)
     (+ t_0 (cbrt (* -0.25 (* (/ h g) (/ h a)))))
     (if (<= g -4.8e-138)
       (fma
        (cbrt (/ 0.5 a))
        (cbrt (- (sqrt (* (+ h g) (- g h))) g))
        (cbrt (* (- (- g) (- g)) (/ 0.5 a))))
       (+ (/ (cbrt (* (* -0.25 (/ h g)) h)) (cbrt a)) t_0)))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt((-g / a));
	double tmp;
	if (g <= -2.15e+151) {
		tmp = t_0 + cbrt((-0.25 * ((h / g) * (h / a))));
	} else if (g <= -4.8e-138) {
		tmp = fma(cbrt((0.5 / a)), cbrt((sqrt(((h + g) * (g - h))) - g)), cbrt(((-g - -g) * (0.5 / a))));
	} else {
		tmp = (cbrt(((-0.25 * (h / g)) * h)) / cbrt(a)) + t_0;
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(-g) / a))
	tmp = 0.0
	if (g <= -2.15e+151)
		tmp = Float64(t_0 + cbrt(Float64(-0.25 * Float64(Float64(h / g) * Float64(h / a)))));
	elseif (g <= -4.8e-138)
		tmp = fma(cbrt(Float64(0.5 / a)), cbrt(Float64(sqrt(Float64(Float64(h + g) * Float64(g - h))) - g)), cbrt(Float64(Float64(Float64(-g) - Float64(-g)) * Float64(0.5 / a))));
	else
		tmp = Float64(Float64(cbrt(Float64(Float64(-0.25 * Float64(h / g)) * h)) / cbrt(a)) + t_0);
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[g, -2.15e+151], N[(t$95$0 + N[Power[N[(-0.25 * N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[g, -4.8e-138], N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Sqrt[N[(N[(h + g), $MachinePrecision] * N[(g - h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[((-g) - (-g)), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(-0.25 * N[(h / g), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if g < -2.14999999999999991e151

   1. Initial program 3.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

      4. lower-neg.f64 3.9

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

   5. Applied rewrites 3.9%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      2. lower-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{h \cdot h}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      4. times-frac N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a}} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \color{blue}{\frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \color{blue}{\left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\color{blue}{\sqrt[3]{\frac{-1}{2}}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      10. lower-cbrt.f64 64.5

          \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{\sqrt[3]{0.5}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   8. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} \]

      3. lower-+.f64 64.5

         \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} \]

   10. Applied rewrites 64.5%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}} \]

   if -2.14999999999999991e151 < g < -4.7999999999999998e-138

   1. Initial program 76.6%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{-2 \cdot a}{g - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. Applied rewrites 97.6%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   7. Taylor expanded in g around -inf

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{-1 \cdot g}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(\mathsf{neg}\left(g\right)\right)}\right)} \]

      2. lower-neg.f64 97.6

         \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(-g\right)}\right)} \]

   9. Applied rewrites 97.6%

      \[\leadsto \frac{1}{\sqrt[3]{\frac{-1}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}} \cdot \sqrt[3]{-2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \color{blue}{\left(-g\right)}\right)} \]

   11. Applied rewrites 97.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}, \sqrt[3]{\left(\left(-g\right) - \left(-g\right)\right) \cdot \frac{0.5}{a}}\right)} \]

   if -4.7999999999999998e-138 < g

   1. Initial program 44.4%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

      4. lower-neg.f64 41.0

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

   5. Applied rewrites 41.0%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

   6. Taylor expanded in g around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      2. lower-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{h \cdot h}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      4. times-frac N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a}} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \color{blue}{\frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \color{blue}{\left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

      9. lower-cbrt.f64 N/A

         \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\color{blue}{\sqrt[3]{\frac{-1}{2}}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

      10. lower-cbrt.f64 70.1

          \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{\sqrt[3]{0.5}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   8. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.1%

       \[\leadsto \sqrt[3]{\frac{h}{a}} \cdot \color{blue}{\sqrt[3]{\frac{h}{g} \cdot -0.25}} + \sqrt[3]{\frac{-g}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 70.4%

       \[\leadsto \frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\color{blue}{\sqrt[3]{a}}} + \sqrt[3]{\frac{-g}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-g}{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (/ (cbrt (* (* -0.25 (/ h g)) h)) (cbrt a)) (cbrt (/ (- g) a))))

C

double code(double g, double h, double a) {
	return (cbrt(((-0.25 * (h / g)) * h)) / cbrt(a)) + cbrt((-g / a));
}

Java

public static double code(double g, double h, double a) {
	return (Math.cbrt(((-0.25 * (h / g)) * h)) / Math.cbrt(a)) + Math.cbrt((-g / a));
}

Julia

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(Float64(-0.25 * Float64(h / g)) * h)) / cbrt(a)) + cbrt(Float64(Float64(-g) / a)))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[N[(N[(-0.25 * N[(h / g), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

   2. mul-1-neg N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

   3. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

   4. lower-neg.f64 26.2

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

5. Applied rewrites 26.2%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

6. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

   2. lower-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   3. unpow2 N/A

      \[\leadsto \sqrt[3]{\frac{\color{blue}{h \cdot h}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   4. times-frac N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   6. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a}} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   7. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \color{blue}{\frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   8. lower-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \color{blue}{\left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

   9. lower-cbrt.f64 N/A

      \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\color{blue}{\sqrt[3]{\frac{-1}{2}}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   10. lower-cbrt.f64 70.2

       \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{\sqrt[3]{0.5}}\right) + \sqrt[3]{\frac{-g}{a}} \]

8. Applied rewrites 70.2%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
9. Step-by-step derivation

10. Applied rewrites 70.2%

    \[\leadsto \sqrt[3]{\frac{h}{a}} \cdot \color{blue}{\sqrt[3]{\frac{h}{g} \cdot -0.25}} + \sqrt[3]{\frac{-g}{a}} \]
11. Step-by-step derivation

12. Applied rewrites 70.4%

    \[\leadsto \frac{\sqrt[3]{\left(-0.25 \cdot \frac{h}{g}\right) \cdot h}}{\color{blue}{\sqrt[3]{a}}} + \sqrt[3]{\frac{-g}{a}} \]
13. Add Preprocessing

Alternative 7: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{h}{a}} \cdot \sqrt[3]{\frac{h}{g} \cdot -0.25} + \sqrt[3]{\frac{-g}{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (* (cbrt (/ h a)) (cbrt (* (/ h g) -0.25))) (cbrt (/ (- g) a))))

C

double code(double g, double h, double a) {
	return (cbrt((h / a)) * cbrt(((h / g) * -0.25))) + cbrt((-g / a));
}

Java

public static double code(double g, double h, double a) {
	return (Math.cbrt((h / a)) * Math.cbrt(((h / g) * -0.25))) + Math.cbrt((-g / a));
}

Julia

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(h / a)) * cbrt(Float64(Float64(h / g) * -0.25))) + cbrt(Float64(Float64(-g) / a)))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[N[(h / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(h / g), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

   2. mul-1-neg N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

   3. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

   4. lower-neg.f64 26.2

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

5. Applied rewrites 26.2%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

6. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

   2. lower-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   3. unpow2 N/A

      \[\leadsto \sqrt[3]{\frac{\color{blue}{h \cdot h}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   4. times-frac N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   6. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a}} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   7. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \color{blue}{\frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   8. lower-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \color{blue}{\left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

   9. lower-cbrt.f64 N/A

      \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\color{blue}{\sqrt[3]{\frac{-1}{2}}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   10. lower-cbrt.f64 70.2

       \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{\sqrt[3]{0.5}}\right) + \sqrt[3]{\frac{-g}{a}} \]

8. Applied rewrites 70.2%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
9. Step-by-step derivation

10. Applied rewrites 70.2%

    \[\leadsto \sqrt[3]{\frac{h}{a}} \cdot \color{blue}{\sqrt[3]{\frac{h}{g} \cdot -0.25}} + \sqrt[3]{\frac{-g}{a}} \]
11. Add Preprocessing

Alternative 8: 15.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(-2 \cdot g\right)} + \sqrt[3]{\frac{-g}{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* (pow (* 2.0 a) -1.0) (* -2.0 g))) (cbrt (/ (- g) a))))

C

double code(double g, double h, double a) {
	return cbrt((pow((2.0 * a), -1.0) * (-2.0 * g))) + cbrt((-g / a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((Math.pow((2.0 * a), -1.0) * (-2.0 * g))) + Math.cbrt((-g / a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64((Float64(2.0 * a) ^ -1.0) * Float64(-2.0 * g))) + cbrt(Float64(Float64(-g) / a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[Power[N[(2.0 * a), $MachinePrecision], -1.0], $MachinePrecision] * N[(-2.0 * g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

   2. mul-1-neg N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

   3. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

   4. lower-neg.f64 26.2

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

5. Applied rewrites 26.2%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

6. Taylor expanded in g around -inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\frac{-g}{a}} \]
7. Step-by-step derivation

   1. lower-*.f64 14.5

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\frac{-g}{a}} \]

8. Applied rewrites 14.5%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\frac{-g}{a}} \]

9. Final simplification 14.5%

   \[\leadsto \sqrt[3]{{\left(2 \cdot a\right)}^{-1} \cdot \left(-2 \cdot g\right)} + \sqrt[3]{\frac{-g}{a}} \]
10. Add Preprocessing

Alternative 9: 74.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (/ (- g) a)) (cbrt (* -0.25 (* (/ h g) (/ h a))))))

C

double code(double g, double h, double a) {
	return cbrt((-g / a)) + cbrt((-0.25 * ((h / g) * (h / a))));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((-g / a)) + Math.cbrt((-0.25 * ((h / g) * (h / a))));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(-g) / a)) + cbrt(Float64(-0.25 * Float64(Float64(h / g) * Float64(h / a)))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(-0.25 * N[(N[(h / g), $MachinePrecision] * N[(h / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

   2. mul-1-neg N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{a}} \]

   3. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{a}}} \]

   4. lower-neg.f64 26.2

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

5. Applied rewrites 26.2%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

6. Taylor expanded in g around inf

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

   2. lower-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{h}^{2}}{a \cdot g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   3. unpow2 N/A

      \[\leadsto \sqrt[3]{\frac{\color{blue}{h \cdot h}}{a \cdot g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   4. times-frac N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a} \cdot \frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   6. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{h}{a}} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   7. lower-/.f64 N/A

      \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \color{blue}{\frac{h}{g}}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   8. lower-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \color{blue}{\left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} + \sqrt[3]{\frac{-g}{a}} \]

   9. lower-cbrt.f64 N/A

      \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\color{blue}{\sqrt[3]{\frac{-1}{2}}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}} \]

   10. lower-cbrt.f64 70.2

       \[\leadsto \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{\sqrt[3]{0.5}}\right) + \sqrt[3]{\frac{-g}{a}} \]

8. Applied rewrites 70.2%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} + \sqrt[3]{\frac{-g}{a}} \]
9. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \sqrt[3]{\frac{-g}{a}}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right)} \]

   3. lower-+.f64 70.2

      \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{h}{a} \cdot \frac{h}{g}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{0.5}\right)} \]

10. Applied rewrites 70.2%

    \[\leadsto \color{blue}{\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{-0.25 \cdot \left(\frac{h}{g} \cdot \frac{h}{a}\right)}} \]
11. Add Preprocessing

Alternative 10: 2.9% accurate, 302.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 0.0)

C

double code(double g, double h, double a) {
	return 0.0;
}

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

public static double code(double g, double h, double a) {
	return 0.0;
}

Python

def code(g, h, a):
	return 0.0

Julia

function code(g, h, a)
	return 0.0
end

MATLAB

function tmp = code(g, h, a)
	tmp = 0.0;
end

Wolfram

code[g_, h_, a_] := 0.0

Derivation

1. Initial program 40.4%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   2. pow1/3 N/A

      \[\leadsto \color{blue}{{\left(\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)\right)}^{\frac{1}{3}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto {\color{blue}{\left(\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)\right)}}^{\frac{1}{3}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   4. unpow-prod-down N/A

      \[\leadsto \color{blue}{{\left(\frac{1}{2 \cdot a}\right)}^{\frac{1}{3}} \cdot {\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}^{\frac{1}{3}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{{\left(\frac{1}{2 \cdot a}\right)}^{\frac{1}{3}} \cdot {\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}^{\frac{1}{3}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. pow1/3 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}}} \cdot {\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}^{\frac{1}{3}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   7. lower-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}}} \cdot {\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}^{\frac{1}{3}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   8. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a}}} \cdot {\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}^{\frac{1}{3}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{\color{blue}{2 \cdot a}}} \cdot {\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}^{\frac{1}{3}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   10. associate-/r* N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}}} \cdot {\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}^{\frac{1}{3}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{2}}}{a}} \cdot {\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}^{\frac{1}{3}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   12. lower-/.f64 N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}}} \cdot {\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}^{\frac{1}{3}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   13. pow1/3 N/A

       \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \color{blue}{\sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   14. lower-cbrt.f64 45.4

       \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \color{blue}{\sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   15. lift-+.f64 N/A

       \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\color{blue}{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   16. +-commutative N/A

       \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\color{blue}{\sqrt{g \cdot g - h \cdot h} + \left(-g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   17. lift-neg.f64 N/A

       \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} + \color{blue}{\left(\mathsf{neg}\left(g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   18. unsub-neg N/A

       \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\color{blue}{\sqrt{g \cdot g - h \cdot h} - g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   19. lower--.f64 45.4

       \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\color{blue}{\sqrt{g \cdot g - h \cdot h} - g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

4. Applied rewrites 45.4%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

5. Taylor expanded in g around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g \cdot \left(1 + {\left(\sqrt{-1}\right)}^{2}\right)}{a}} \cdot \sqrt[3]{\frac{1}{2}}\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt[3]{\frac{g \cdot \left(1 + {\left(\sqrt{-1}\right)}^{2}\right)}{a}} \cdot \sqrt[3]{\frac{1}{2}}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\sqrt[3]{\frac{g \cdot \left(1 + {\left(\sqrt{-1}\right)}^{2}\right)}{a}} \cdot \sqrt[3]{\frac{1}{2}}} \]

   3. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\sqrt[3]{\frac{g \cdot \left(1 + {\left(\sqrt{-1}\right)}^{2}\right)}{a}} \cdot \sqrt[3]{\frac{1}{2}}} \]

   4. lower-cbrt.f64 N/A

      \[\leadsto -\color{blue}{\sqrt[3]{\frac{g \cdot \left(1 + {\left(\sqrt{-1}\right)}^{2}\right)}{a}}} \cdot \sqrt[3]{\frac{1}{2}} \]

   5. lower-/.f64 N/A

      \[\leadsto -\sqrt[3]{\color{blue}{\frac{g \cdot \left(1 + {\left(\sqrt{-1}\right)}^{2}\right)}{a}}} \cdot \sqrt[3]{\frac{1}{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto -\sqrt[3]{\frac{\color{blue}{g \cdot \left(1 + {\left(\sqrt{-1}\right)}^{2}\right)}}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   7. +-commutative N/A

      \[\leadsto -\sqrt[3]{\frac{g \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} + 1\right)}}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   8. unpow2 N/A

      \[\leadsto -\sqrt[3]{\frac{g \cdot \left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + 1\right)}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   9. rem-square-sqrt N/A

      \[\leadsto -\sqrt[3]{\frac{g \cdot \left(\color{blue}{-1} + 1\right)}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   10. metadata-eval N/A

       \[\leadsto -\sqrt[3]{\frac{g \cdot \color{blue}{0}}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   11. lower-cbrt.f64 3.0

       \[\leadsto -\sqrt[3]{\frac{g \cdot 0}{a}} \cdot \color{blue}{\sqrt[3]{0.5}} \]

7. Applied rewrites 3.0%

   \[\leadsto \color{blue}{-\sqrt[3]{\frac{g \cdot 0}{a}} \cdot \sqrt[3]{0.5}} \]
8. Step-by-step derivation

9. Applied rewrites 3.0%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024309 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))